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<meta property="og:description" content="说在前面这个问题我还没有解决. 题目已知平面向量 $v_1,v_2,v_3,\cdots,v_n$ 满足 $\sum_{i&#x3D;1}^n|\vec{v_i}|&#x3D;1$ . 若必存在若干个向量, 它们的和的模不小于 $k$ , 求 $k$ 的最大值. 原题如下  给出的解法虽然很精彩, 但是并不是最大值 (最少是没证明 $1&#x2F;4$ 是最大值) . 于是就产生了这个求最大值的想法. 猜想我猜测结果是 $1&#x2F;">
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id="post-info"><h1 class="post-title">一个问题</h1><div id="post-meta"><div class="meta-firstline"><span class="post-meta-date"><i class="far fa-calendar-alt fa-fw post-meta-icon"></i><span class="post-meta-label">发表于</span><time class="post-meta-date-created" datetime="2021-05-01T06:50:29.000Z" title="发表于 2021-05-01 14:50:29">2021-05-01</time><span class="post-meta-separator">|</span><i class="fas fa-history fa-fw post-meta-icon"></i><span class="post-meta-label">更新于</span><time class="post-meta-date-updated" datetime="2021-08-14T04:39:34.402Z" title="更新于 2021-08-14 12:39:34">2021-08-14</time></span><span class="post-meta-categories"><span class="post-meta-separator">|</span><i class="fas fa-inbox fa-fw post-meta-icon"></i><a class="post-meta-categories" href="/categories/%E6%95%B0%E5%AD%A6/">数学</a><i class="fas fa-angle-right post-meta-separator"></i><i class="fas fa-inbox fa-fw post-meta-icon"></i><a class="post-meta-categories" href="/categories/%E6%95%B0%E5%AD%A6/%E5%85%B6%E4%BB%96/">其他</a></span></div><div class="meta-secondline"><span class="post-meta-separator">|</span><span class="post-meta-pv-cv"><i class="far fa-eye fa-fw post-meta-icon"></i><span class="post-meta-label">阅读量:</span><span id="busuanzi_value_page_pv"></span></span><span class="post-meta-separator">|</span><span class="post-meta-commentcount"><i class="far fa-comments fa-fw post-meta-icon"></i><span class="post-meta-label">评论数:</span><a href="/math/other/a_problem/#post-comment"><span id="twikoo-count"></span></a></span></div></div></div></header><main class="layout" id="content-inner"><div id="post"><article class="post-content" id="article-container"><h1 id="说在前面"><a href="#说在前面" class="headerlink" title="说在前面"></a>说在前面</h1><p>这个问题我还没有解决.</p>
<h1 id="题目"><a href="#题目" class="headerlink" title="题目"></a>题目</h1><p>已知平面向量 $v_1,v_2,v_3,\cdots,v_n$ 满足 $\sum_{i=1}^n|\vec{v_i}|=1$ . 若必存在若干个向量, 它们的和的模不小于 $k$ , 求 $k$ 的最大值.</p>
<p>原题如下</p>
<p><img src="1.jpg"></p>
<p>给出的解法虽然很精彩, 但是并不是最大值 (最少是没证明 $1/4$ 是最大值) . 于是就产生了这个求最大值的想法.</p>
<h1 id="猜想"><a href="#猜想" class="headerlink" title="猜想"></a>猜想</h1><p>我猜测结果是 $1/\pi$ .</p>
<h1 id="一些结果"><a href="#一些结果" class="headerlink" title="一些结果"></a>一些结果</h1><p>令 $P\subseteq {1,2,3,\cdots,n}$ , 且满足 $\left|\sum_{i\in P}v_i\right|$ 为所有向量相加的组合中能取到的最大值, 我们来考虑如何选取得到这个向量的集合. 设一个向量 $\vec{w}\not=\vec 0$ , 如何取向量并相加才能使得这个相加后的向量在 $\vec{w}$ 上的投影的模最大? 很显然, 取一条过原点的直线, 然后分别将直线的两边的所有向量相加 (在直线上的向量相加与否并不影响这个模的取值), 得到的两个向量必有一个在 $\vec{w}$ 上的投影的模是最大的, 但是这样取到的最大的模必然小于等于 $\left|\sum_{i\in P}v_i\right|$ (当 $\vec w$ 与 $\sum_{i\in P}v_i$ 的方向相同或相反时取等号). 于是, 向量 $\left|\sum_{i\in P}v_i\right|$ 必然是某条过原点的直线的一边的所有向量的和的模.</p>
<p><img src="2.png"></p>
<p>因此, 我们就可以利用这个结果来设计一个较低复杂度 $O(n)$ 的程序, 来使我的猜想更加可信.</p>
<h1 id="代码验证"><a href="#代码验证" class="headerlink" title="代码验证"></a>代码验证</h1><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line">rand = <span class="keyword">lambda</span> num, t: np.random.rand(num, t) - <span class="number">0.5</span></span><br><span class="line">sin = np.sin</span><br><span class="line">cos = np.cos</span><br><span class="line">arcsin = np.arcsin</span><br><span class="line">arccos = np.arccos</span><br><span class="line">sqrt = np.sqrt</span><br><span class="line">pi = np.pi</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">angle</span>(<span class="params">x0, y0</span>):</span></span><br><span class="line">    x = x0 / sqrt(x0**<span class="number">2</span> + y0**<span class="number">2</span>)</span><br><span class="line">    y = y0 / sqrt(x0**<span class="number">2</span> + y0**<span class="number">2</span>)</span><br><span class="line">    efactor = <span class="number">0.01</span></span><br><span class="line">    <span class="keyword">if</span> y &gt; <span class="number">0</span>:</span><br><span class="line">        theta = arcsin(y)</span><br><span class="line">        <span class="keyword">if</span> cos(theta) - x &lt; efactor:</span><br><span class="line">            <span class="keyword">return</span> theta</span><br><span class="line">        <span class="keyword">else</span>:</span><br><span class="line">            <span class="keyword">return</span> pi - theta</span><br><span class="line">    <span class="keyword">else</span>:</span><br><span class="line">        theta = arcsin(y) + <span class="number">2</span>*pi</span><br><span class="line">        <span class="keyword">if</span> cos(theta) - x &lt; efactor:</span><br><span class="line">            <span class="keyword">return</span> theta</span><br><span class="line">        <span class="keyword">else</span>:</span><br><span class="line">            <span class="keyword">return</span> <span class="number">3</span>*pi - theta</span><br><span class="line">        </span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">angle</span>(<span class="params">x0, y0</span>):</span></span><br><span class="line">    x = x0 / sqrt(x0**<span class="number">2</span> + y0**<span class="number">2</span>)</span><br><span class="line">    y = y0 / sqrt(x0**<span class="number">2</span> + y0**<span class="number">2</span>)</span><br><span class="line">    efactor = <span class="number">0.01</span></span><br><span class="line">    <span class="keyword">if</span> y &gt; <span class="number">0</span>:</span><br><span class="line">        theta = arcsin(y)</span><br><span class="line">        <span class="keyword">if</span> cos(theta) - x &lt; efactor:</span><br><span class="line">            <span class="keyword">return</span> theta</span><br><span class="line">        <span class="keyword">else</span>:</span><br><span class="line">            <span class="keyword">return</span> pi - theta</span><br><span class="line">    <span class="keyword">else</span>:</span><br><span class="line">        theta = arcsin(y) + <span class="number">2</span>*pi</span><br><span class="line">        <span class="keyword">if</span> cos(theta) - x &lt; efactor:</span><br><span class="line">            <span class="keyword">return</span> theta</span><br><span class="line">        <span class="keyword">else</span>:</span><br><span class="line">            <span class="keyword">return</span> <span class="number">3</span>*pi - theta</span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">rand_vec</span>(<span class="params">num</span>):</span></span><br><span class="line">    res = []</span><br><span class="line">    <span class="keyword">for</span> x, y <span class="keyword">in</span> rand(num, <span class="number">2</span>):</span><br><span class="line">        res.append(Vector(x, y))</span><br><span class="line">    <span class="keyword">return</span> res</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">init_vec</span>(<span class="params">num</span>):</span></span><br><span class="line">    vec_arr = rand_vec(num)</span><br><span class="line">    res = []</span><br><span class="line">    length = <span class="number">0</span></span><br><span class="line">    <span class="keyword">for</span> vec <span class="keyword">in</span> vec_arr:</span><br><span class="line">        length += vec.length()</span><br><span class="line">    <span class="keyword">for</span> vec <span class="keyword">in</span> vec_arr:</span><br><span class="line">        res.append(Vector(vec.x / length, vec.y / length))</span><br><span class="line">    <span class="keyword">return</span> res</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">init_sum_vec_list</span>(<span class="params">MAXNUM = <span class="number">100</span></span>):</span></span><br><span class="line">    vec_list = init_vec(MAXNUM)</span><br><span class="line">    vec_list.sort(key=(<span class="keyword">lambda</span> vec: vec.angle))</span><br><span class="line">    sum_vec = vec_list[-<span class="number">1</span>]</span><br><span class="line">    sum_vec_list = []</span><br><span class="line">    bedex = <span class="number">0</span></span><br><span class="line">    edex = <span class="number">0</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">for</span> bdex <span class="keyword">in</span> <span class="built_in">range</span>(MAXNUM):</span><br><span class="line">        sum_vec -= vec_list[bdex - <span class="number">1</span>]</span><br><span class="line">        bedex = edex</span><br><span class="line">        <span class="keyword">for</span> edex <span class="keyword">in</span> <span class="built_in">range</span>(bedex, bedex + MAXNUM):</span><br><span class="line">            sum_vec += vec_list[edex % MAXNUM]</span><br><span class="line">            <span class="keyword">if</span> edex + <span class="number">1</span> &lt; MAXNUM:</span><br><span class="line">                <span class="keyword">if</span> vec_list[(edex + <span class="number">1</span>) % MAXNUM].angle - vec_list[bdex % MAXNUM].angle &gt; pi:</span><br><span class="line">                    sum_vec_list.append(sum_vec)</span><br><span class="line">                    <span class="keyword">break</span></span><br><span class="line">            <span class="keyword">else</span>:</span><br><span class="line">                <span class="keyword">if</span> vec_list[(edex + <span class="number">1</span>) % MAXNUM].angle - vec_list[bdex % MAXNUM].angle &gt; -pi:</span><br><span class="line">                    sum_vec_list.append(sum_vec)</span><br><span class="line">                    <span class="keyword">break</span></span><br><span class="line">    <span class="keyword">return</span> sum_vec_list</span><br><span class="line"></span><br><span class="line">res_len = []</span><br><span class="line"><span class="keyword">for</span> i <span class="keyword">in</span> <span class="built_in">range</span>(<span class="number">100</span>):</span><br><span class="line">    res_len.append(<span class="built_in">max</span>(init_sum_vec_list()).length())</span><br><span class="line">print(<span class="built_in">min</span>(res_len))</span><br></pre></td></tr></table></figure>

<p>代码没有注释是因为<del>优秀的代码不需要注释</del>我太懒了. 最后结果算出来基本上都大于 0.5, 想来可能是简单的等概率随机分布并不适合生成随机向量, 以后有时间再优化一下吧.</p>
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